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Mirrors > Home > MPE Home > Th. List > mvrcl | Structured version Visualization version GIF version |
Description: A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mvrcl.s | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mvrcl.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
mvrcl.b | ⊢ 𝐵 = (Base‘𝑃) |
mvrcl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mvrcl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mvrcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
mvrcl | ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | mvrcl.v | . . 3 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
3 | eqid 2728 | . . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
4 | mvrcl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | mvrcl.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
6 | mvrcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
7 | 1, 2, 3, 4, 5, 6 | mvrcl2 21929 | . 2 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘(𝐼 mPwSer 𝑅))) |
8 | fvexd 6912 | . . 3 ⊢ (𝜑 → (𝑉‘𝑋) ∈ V) | |
9 | eqid 2728 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | eqid 2728 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
11 | 1, 9, 10, 3, 7 | psrelbas 21879 | . . . 4 ⊢ (𝜑 → (𝑉‘𝑋):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
12 | 11 | ffund 6726 | . . 3 ⊢ (𝜑 → Fun (𝑉‘𝑋)) |
13 | fvexd 6912 | . . 3 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
14 | snfi 9069 | . . . 4 ⊢ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin) |
16 | eqid 2728 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | eqid 2728 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
18 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝐼 ∈ 𝑊) |
19 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑅 ∈ Ring) |
20 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑋 ∈ 𝐼) |
21 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) | |
22 | eldifsn 4791 | . . . . . . . 8 ⊢ (𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ↔ (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) | |
23 | 21, 22 | sylib 217 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) |
24 | 23 | simpld 494 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) |
25 | 2, 10, 16, 17, 18, 19, 20, 24 | mvrval2 21925 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ((𝑉‘𝑋)‘𝑥) = if(𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅))) |
26 | 23 | simprd 495 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
27 | 26 | neneqd 2942 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ¬ 𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
28 | 27 | iffalsed 4540 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → if(𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) |
29 | 25, 28 | eqtrd 2768 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ((𝑉‘𝑋)‘𝑥) = (0g‘𝑅)) |
30 | 11, 29 | suppss 8199 | . . 3 ⊢ (𝜑 → ((𝑉‘𝑋) supp (0g‘𝑅)) ⊆ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) |
31 | suppssfifsupp 9404 | . . 3 ⊢ ((((𝑉‘𝑋) ∈ V ∧ Fun (𝑉‘𝑋) ∧ (0g‘𝑅) ∈ V) ∧ ({(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin ∧ ((𝑉‘𝑋) supp (0g‘𝑅)) ⊆ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑉‘𝑋) finSupp (0g‘𝑅)) | |
32 | 8, 12, 13, 15, 30, 31 | syl32anc 1376 | . 2 ⊢ (𝜑 → (𝑉‘𝑋) finSupp (0g‘𝑅)) |
33 | mvrcl.s | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
34 | mvrcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
35 | 33, 1, 3, 16, 34 | mplelbas 21933 | . 2 ⊢ ((𝑉‘𝑋) ∈ 𝐵 ↔ ((𝑉‘𝑋) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑉‘𝑋) finSupp (0g‘𝑅))) |
36 | 7, 32, 35 | sylanbrc 582 | 1 ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 {crab 3429 Vcvv 3471 ∖ cdif 3944 ⊆ wss 3947 ifcif 4529 {csn 4629 class class class wbr 5148 ↦ cmpt 5231 ◡ccnv 5677 “ cima 5681 Fun wfun 6542 ‘cfv 6548 (class class class)co 7420 supp csupp 8165 ↑m cmap 8845 Fincfn 8964 finSupp cfsupp 9386 0cc0 11139 1c1 11140 ℕcn 12243 ℕ0cn0 12503 Basecbs 17180 0gc0g 17421 1rcur 20121 Ringcrg 20173 mPwSer cmps 21837 mVar cmvr 21838 mPoly cmpl 21839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-tset 17252 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-mgp 20075 df-ur 20122 df-ring 20175 df-psr 21842 df-mvr 21843 df-mpl 21844 |
This theorem is referenced by: mvrf2 21935 subrgmvrf 21972 mplcoe3 21976 mplcoe5lem 21977 mplcoe5 21978 mplcoe2 21979 mplbas2 21980 evlsvarpw 22040 mpfproj 22048 mpfind 22053 mhpvarcl 22072 vr1cl 22136 evlsvarval 41798 selvcllem5 41815 selvvvval 41818 |
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